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Existence of positive solutions to $$p$$-Kirchhoff-type problem without compactness conditions. (English) Zbl 1317.35058
Summary: In this paper, we study the existence of positive solutions to the $$p$$-Kirchhoff-type problem $\begin{cases} \Bigg(a+\lambda\Bigg(\int_{\mathbb R^N}(|\nabla u|^p+b|u|^p)dx\Bigg)^\tau\Bigg)(-\Delta_pu+b|u|^{p-2}u)\\ =|u|^{m-2}u+\mu|u|^{q-2}u,\;x\in\mathbb R^N,\\ u(x)>0,\;x\in\mathbb R^N,\;u(x)\in W^{1,p}(\mathbb R^N), \end{cases} \eqno{(0.1)}$ where $$a,b>0$$,$${\tau},{\lambda}\geq 0,{\mu}\in \mathbb R$$, $$1<p<N,p<q<m<p^\ast =\frac{pN}{N-p}$$. A new existence result for (0.1) is obtained by the Nehari manifold method. This result can be regarded as an extension of the result in [Y. Li et al., J. Differ. Equations 253, No. 7, 2285–2294 (2012; Zbl 1259.35078)].

##### MSC:
 35J40 Boundary value problems for higher-order elliptic equations 35B09 Positive solutions to PDEs
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##### References:
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